Abstract
The main purpose of this paper is to investigate the converse problems of some well-known results related to the generalized Drazin (g-Drazin for short) inverse in Banach algebras. Let \({\mathcal {A}}\) be a Banach algebra and \(a,b\in {\mathcal {A}}\). First, we give the relationship between the Drazin (g-Drazin, group) invertibility of a, b and that of the sum \(a+b\) under certain conditions. Then, for a given polynomial \(f(x)\in {\mathbb {C}}[x]\), the g-Drazin invertibility of f(a), \(f(a^{d})\), f(ab), \(f(1-ab)\) and \(f(a+b)\) are investigated.
Similar content being viewed by others
Data availability
The Date is applicable.
References
Ben-Isreal, A., Greville, T.N.E.: Generalized Inverse: Theory and Applications, 2nd edn. Springer, New York (2003)
Castro-González, N., Koliha, J.J.: New additive results for the g-Drazin inverse. Proc. R. Soc. Edinburgh Sect. A 134, 1085–1097 (2004)
Chen, H.Y., Sheibani, M.: The g-Drazin inverse of the sum in Banach algebras. Linear Multilinear Algebra 70, 53–65 (2022)
Chen, X.Y., Wang, Q.W.: The \(\eta \)-(anti-)Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation. Banach J. Math. Anal. 17, 40 (2023)
Chen, Y., Wang, Q.W., Xie, L.M.: Dual quaternion matrix equation \(AXB = C\) with applications. Symmetry 16, 287 (2024)
Chen, J.L., Zhuang, G.F., Wei, Y.M.: The Drazin inverse of a sum of morphisms. Acta. Math. Sci. 29A, 538–552 (2009). ((in Chinese))
Cvetković-Ilić, D.S.: The generalized Drazin inverse with commutativity up to a factor in a Banach algebra. Linear Algebra Appl. 431, 783–791 (2009)
Cvetković-Ilić, D.S.: Some results on the (2, 2, 0) Drazin inverse problem. Linear Algebra Appl. 438, 4726–4741 (2013)
Cvetković-Ilić, D.S., Liu, X.J., Wei, Y.M.: Some additive results for the generalized Drazin inverse in a Banach algebra. Electron. J. Linear Algebra 22, 1049–1058 (2011)
Cvetković-Ilić, D.S., Wei, Y.M.: Algebraic Properties of Generalized Inverses. Springer, New York (2017)
Deng, C.Y.: The Drazin inverse of bounded operators with commutativity up to a factor. Appl. Math. Comput. 206, 695–703 (2008)
Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Monthly 65, 506–514 (1958)
Hartwig, R.E., Luh, J.: A note on the group structure of unit regular ring elements. Pacific J. Math. 71, 449–461 (1977)
Hartwig, R.E., Shoaf, J.: Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices. J. Aust. Math. Soc. 24, 10–34 (1977)
Hartwig, R.E., Wang, G.R., Wei, Y.M.: Some additive results on Drazin inverse. Linear Algebra Appl. 322, 207–217 (2001)
Koliha, J.J.: A generalized Drazin inverse. Glasgow Math. J. 38, 367–381 (1996)
Liao, Y.H., Chen, J.L., Cui, J.: Cline’s formula for the generalized Drazin inverse. Bull. Malays. Math. Sci. Soc. 37, 37–42 (2014)
Ljubisavljević, J., Cvetković-Ilić, D.S.: Additive results for the Drazin inverse of block matrices and applications. J. Comput. Appl. Math. 235, 3683–3690 (2011)
Mosić, D.: A note on Cline’s formula for the generalized Drazin inverse. Linear Multilinear Algebra 63, 1106–1110 (2015)
Mosić, D.: Additive results for the generalized Drazin inverse in a Banach algebra. Bull. Malays. Math. Sci. Soc. 40, 1465–1478 (2017)
Višnjić, J.: On additive properties of the Drazin inverse of block matrices and representations. Appl. Math. Comput. 250, 444–450 (2015)
Xu, X.L., Wang, Q.W.: The consistency and the general common solution to some quaternion matrix equations. Ann. Funct. Anal. 14, 53 (2023)
Zhang, Y., Wang, Q.W., Xie, L.M.: The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations. Symmetry 16, 361 (2024)
Zhuang, G.F., Chen, J.L., Cui, J.: Jacobson’s Lemma for the generalized Drazin inverse. Linear Algebra Appl. 436, 742–746 (2012)
Acknowledgements
The author would like to thank the referees for their helpful suggestions for the improvement of this paper. This research was supported by talent introduction project of Zhejiang Shuren University (No. 2023R025), Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (No. SX202202), China Postdoctoral Science Foundation (No. 2020M671281), the National Natural Science Foundation of China (No.11871145), NSF of Jiangsu Province (No. BK20200944).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Qing-Wen Wang.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zou, H. Some converse problems on the g-Drazin invertibility in Banach algebras. Ann. Funct. Anal. 15, 41 (2024). https://doi.org/10.1007/s43034-024-00344-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43034-024-00344-x